Introduction

The terahertz (THz) frequency belongs to the region of electromagnetic spectrum in between the wavelengths 1,000 and 30 μm (i.e. 0.3–10 THz). Terahertz sources are in great demand nowadays for various application areas such as spectroscopy (Grischkowsky et al. 1990), bio-sensing (Debus and Bolivar 2007), quality inspection in various industrial branches (Yasui et al. 2005; Stoik et al. 2008; Jördens and Koch 2008), medical and pharmaceutical applications (Fitzgerald et al. 2005; Siegel 2004), THz astronomy (Siegel 2007), etc. The inherent advantages and potential benefits of the THz frequency regime (0.3–10 THz) for both civilian and defense applications have generated lot of research interest in the development of THz components, sources and detectors. Though the THz region has several applications, lack of suitable THz sources has hindered the developments of THz systems. Most of the available THz sources are complex, bulky and not operational at room temperature except some recently developed THz sources based on cascaded frequency multipliers which are frequency tunable, operational at room temperature and have excellent signal quality. Recently reported 1.9–2.7 THz sources based on cascaded frequency multipliers are capable of generating 10–2 μW of power and these are useful as local oscillators for astronomical receivers (Ward et al. 2004; Heyminck et al. 2009; Crowe et al. 2011a, b; Maestrini et al. 2012). Thus, the power delivering capabilities of these sources are not sufficient to use those as frequency generators in powerful THz transmitting systems.

Among two-terminal solid-state sources impact avalanche transit time (IMPATT) devices have already emerged as high-power, high-efficiency solid-state sources for both microwave (3–30 GHz) and millimeter-wave (30–300 GHz) frequency bands (Midford and Bernick 1979; Chang et al. 1977; Gray et al. 1969). Si and GaAs IMPATTs are already established as powerful and efficient sources at different millimeter-wave window frequencies (Luy et al. 1987; Dalle et al. 1990; Luschas et al. 2002a, b; Shih et al. 1983; Eisele and Haddad 1995). Eisele et al. (1996) first experimentally demonstrated the InP-based IMPATT operation at W-band. But so far as authors’ knowledge is concerned, the experimental results on THz performance of InP IMPATTs are not available in the published literatures. But some recent theoretical study reveals the potentiality of InP-based IMPATTs as THz sources (Mukherjee et al. 2007, 2010). In recent years the potentiality of wide bandgap materials (SiC, GaN) for generation of RF power at THz frequencies has been reported (Mukherjee and Mazumder 2007; Mukherjee et al. 2009; Panda et al. 2007, 2001; Banerjee et al. 2010). Yuan et al. (2001) first demonstrated the 4H-SiC based p+–n–n−–n+ structured IMPATT diode oscillator operating at X-band. Later, Vassilevski et al. (2001) experimentally obtained better RF power output from single-drift region (SDR) p+–n–n+ 4H-SiC IMPATT diode oscillator. So far as authors’ knowledge is concerned, experimental report on GaN-based IMPATT sources is not reported so far in the published literatures. This may be due to lack of mature fabrication technology of SiC and GaN-based IMPATT devices at mm-wave and THz frequencies.

In the present paper, the authors have studied the millimeter-wave and THz performance of GaAs, Si, InP, 4H-SiC and Wz-GaN-based DDR IMPATTs using an avalanche response time-based analysis (Acharyya and Banerjee 2012, 2013; Acharyya et al. 2012) following a double-iterative field maximum simulation method (Roy et al. 1979, 1985) based on Gummel–Blue model (Gummel and Blue 1967) to obtain the upper cut-off frequencies for IMPATT operation. The high-frequency performance of IMPATT devices depends on the material parameters such as carrier ionization rates (αn, αp), saturated drift velocities (vsn, vsp), etc. of the base semiconductors. The avalanche response time of the charge carriers (electrons and holes) is related to the avalanche multiplication process and plays an important role in determining the high-frequency performance of the device. Lower avalanche response time facilitates higher frequency of operation. Thus, the avalanche response times associated with GaAs, Si, InP, 4H-SiC and Wz-GaN based DDR IMPATTs are calculated at different mm-wave and THz frequencies to determine the upper cut-off frequency limits of those devices. DDR structure is preferred over SDR structure for investigation because the presence of two drift layers (n- and p-layers) in DDR structure leads to higher drift zone voltage at a particular frequency of operation as compared with SDR structure in which only one drift layer is available (n- or p-drift layer); and this ensures higher DC to RF conversion efficiency (η) and higher RF power output (PRF) (Sze and Ng 2010). The simulation obtained results are compared with the experimental results to validate the DC and high-frequency simulation technique adopted by the authors to study the mm-wave and THz performance of the DDR IMPATTs based on different semiconductors.

Material parameters and design method

Material parameters

The electric field variation of carrier ionization rates in GaAs, Si, InP and Wz-GaN is given by

$$ \alpha_{{{\text{n}} . {\text{p}}}} \left( \xi \right) = A_{\text{n,p}} \exp \left[ {\left( {\frac{{ - B_{{{\text{n}} . {\text{p}}}} }}{\xi }} \right)^{m} } \right] $$
(1)

where the value of the constant m = 1 for Si, InP and Wz-GaN and m = 2 for GaAs. The values of the ionization coefficients An,p and Bn.p for a wide field range in GaAs for <100> orientation have been taken from experimental data of Ito et al. (1978). The values of An,p and Bn.p for Si have been taken from the experimental results reported by Grant (1973). The ionization coefficients An,p and Bn.p in <100> InP have been obtained from the experimental data of Kao and Crowell (1980) for the lower field range (2.5 × 107 to 5.0 × 107 V m−1) and Umebu et al. (1980) for the higher field range (5.0 × 107 to 8.0 × 107 V m−1). The same coefficients (An,p and Bn.p) of Wz-GaN are taken from recently published report of Kunihiro et al. (1999). The field variation of carrier ionization rates in 4H-SiC is given by

$$ \alpha_{\text{n}} \left( \xi \right) = \left( {\frac{\xi }{{A_{\text{n}} }}} \right)\exp \left( {\frac{{ - B_{\text{n}} }}{{\xi^{2} }}} \right), $$
(2)
$$ \alpha_{\text{p}} \left( \xi \right) = \left( {\frac{\xi }{7}} \right)\exp \left[ {\frac{ - 1}{{\left( {A_{\text{p}} \xi^{2} + B_{\text{p}} \xi } \right)}}} \right]. $$
(3)

The values of the constants An,p and Bn.p for a wide field range in 4H-SiC are taken from experimentally reported data (Konstantinov et al. 1997). The electron and hole drift velocity versus field characteristics of Si have an experimental field dependence given by

$$ v_{\text{n,p}} \left( \xi \right) = v_{\text{sn,sp}} \left[ {1 - \exp \left( {\frac{{ - \mu_{\text{n,p}} \xi }}{{v_{\text{sn,sp}} }}} \right)} \right]. $$
(4)

The values of vsn and vsp are taken from the experimental data of Canali et al. (1971). The negative differential mobility in the electron drift velocity versus electric field characteristics of group III-IV semiconductors GaAs (Kramer and Micrea 1975), InP (Kramer and Micrea 1975) and Wz-GaN (Shiyu and Wang 2008) have been taken into account in the computer simulation through the expression

$$ v_{\text{n}} \left( \xi \right) = \frac{{\left[ {\mu_{\text{n}} \xi + v_{\text{sn}} \left( {{\xi \mathord{\left/ {\vphantom {\xi {\xi_{\text{c}} }}} \right. \kern-\nulldelimiterspace} {\xi_{\text{c}} }}} \right)^{4} } \right]}}{{\left[ {1 + \left( {{\xi \mathord{\left/ {\vphantom {\xi {\xi_{\text{c}} }}} \right. \kern-\nulldelimiterspace} {\xi_{\text{c}} }}} \right)^{4} } \right]}} $$
(5)

which incorporates a peak in the drift velocity at low field (ξc) followed by a velocity saturation at high electric field. The hole drift velocity versus field characteristics of GaAs, InP and Wz-GaN are same as Si as given by Eq. (4). The field dependence of electron and hole drift velocity in 4H-SiC are given by (Vassilevski et al. 2000):

$$ v_{\text{n,p}} \left( \xi \right) = \left[ {\frac{{\mu_{\text{n,p}} \xi }}{{\left( {1 + \left( {{{\mu_{\text{n,p}} \xi } \mathord{\left/ {\vphantom {{\mu_{\text{n,p}} \xi } {v_{\text{sn,sp}} }}} \right. \kern-\nulldelimiterspace} {v_{\text{sn,sp}} }}} \right)^{\kappa } } \right)}}} \right]^{{{1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }}} , $$
(6)

where the value of κ = 1.20. All other material parameters such as bandgap (Eg), intrinsic carrier concentration (ni), effective density of states of conduction and valance bands (Nc, Nv), diffusion coefficients (Dn, Dp), mobilities (μn, μp) and diffusion lengths (Ln, Lp) of charge carriers and permittivity (εs) of the semiconductor materials under consideration are taken from recently published reports (Electronic Archive 2012).

Design method

The frequency of operation of an IMPATT diode essentially depends on the transit time (τT) of charge carriers to cross the depletion layer of the device. IMPATT diodes having double-drift region n+–n–p–p+ structure are first designed for a particular frequency (fd) by the transit time formula of Sze and Ryder (1971) given by Wn,p = 0.37 vsn,sp/fd, where Wn,p and vsn,sp are the n- or p-side depletion layer widths, saturation velocities of electrons/holes respectively. Here n+- and p+-layers are highly doped layers whose doping concentrations are taken to be Nn+ = Np+ = 1026 m−3. The background doping concentrations of n- and p-depletion regions (ND, NA) are initially chosen according to the design frequency. The electric field profile using the above doping profile is obtained from the DC simulation described in next subsection (Roy et al. 1979). The input doping profile is adjusted so that the electric field just punches through the depletion layers (Wn,p) for a particular value of fd and a particular biasing current density (J0). The high-frequency computer simulation described elsewhere (Roy et al. 1985) based on Gummel–Blue approach (Gummel and Blue 1967) is then carried out to obtain the admittance characteristics of the device. The peak optimum frequency (fp) corresponding to the peak negative conductance is determined from the admittance characteristics. If the magnitude of fp differs very much from fd, the value of J0 is varied and the computer simulation program is run till the value of fp is nearly equal to the value of fd. The bias current density is thus fixed for the particular design frequency. Realistic doping profile for flat profile diode has been used for the present analysis. The doping profile at the interfaces of epitaxy and substrate (n+n-interface) is approximated as error function. The doping profile near the p+p-interface has been made realistic by suitable exponential functions. Structural and doping parameters of the designed DDR IMPATTs based on GaAs, Si, InP, 4H-SiC and Wz-GaN are listed in Table 1.

Table 1 Structural and doping parameters
Full size table

Simulation technique

One-dimensional model of reverse biased n+–n–p–p+ lateral DDR IMPATT structure, shown in Fig. 1, is used to simulate the DC and high-frequency properties of the device under optical illumination. The physical phenomena take place in the semiconductor bulk along the symmetry axis of the DDR IMPATT devices. Thus, the one-dimensional model of IMPATT devices considered in this work is justified.

Fig. 1
figure 1

One-dimensional model of DDR IMPATT device

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DC simulation

The DC electric field and normalized current density profiles in the depletion layer of the device are obtained from simultaneous numerical solution of fundamental device equations such as Poisson’s equation, combined carrier continuity equation, current density equations and mobile space charge equation subject to appropriate boundary conditions (Acharyya and Banerjee 2012). A double-iterative field maximum simulation method based on Gummel–Blue approach (Gummel and Blue 1967) described elsewhere (Roy et al. 1979; Acharyya and Banerjee 2012) is used to solve these equations and obtain the electric field and normalized current density profiles. In the above-mentioned simulation method, the computation starts from the field maximum near the metallurgical junction. The boundary conditions for the electric field (ξ(x)) at the depletion layer edges are given by

$$ \xi \left( { - x_{1} } \right) = 0\quad {\text{and}}\quad \xi \left( {x_{2} } \right) = 0 $$
(7)

Similarly, the boundary conditions for normalized difference of hole and electron current density, P(x) = (Jp(x) − Jn(x))/J0 (where J0 = Jp + Jn) at the depletion layer edges, i.e., at x = −x1 and x = x2 are given by

$$ P\left( { - x_{1} } \right) = \left( {\frac{2}{{M_{\text{p}} }} - 1} \right)\quad {\text{and}}\quad P\left( {x_{2} } \right) = \left( {1 - \frac{2}{{M_{\text{n}} }}} \right), $$
(8)

where Mn and Mp are the electron and hole multiplication factors whose values are infinitely large of the order of ~106 under dark or un-illuminated condition of the device. Electric field profiles (ξ(x) vs. x) and normalized current profiles (P(x) vs. x) are obtained from the DC simulation as mentioned earlier. The breakdown voltage (VB) and the avalanche zone voltage drop (VA) are calculated by integrating the spatial field profile (ξ(x) vs. x) over the total depletion layer width (W) and by integrating the spatial field profile over the total avalanche layer width (xA), respectively, i.e.,

$$ V_{\text{B}} = \int\limits_{{ - x_{1} }}^{{x_{2} }} {\xi \left( x \right)dx} \quad {\text{and}}\quad V_{\text{A}} = \int\limits_{{ - x_{{{\text{A}}_{1} }} }}^{{x_{{{\text{A}}_{2} }} }} {\xi \left( x \right)dx} . $$
(9)

The DC to RF conversion efficiency is calculated from the semi quantitative formula (Scharfetter and Gummel 1969):

$$ \eta \left( \% \right) = \frac{{2m_{\text{p}} }}{\pi } \times \frac{{V_{\text{D}} }}{{V_{\text{B}} }} $$
(10)

where VD = (VB − VA) = voltage drop across the drift region and mp = 1/2.

Calculation of avalanche response time

The avalanche response time of DDR IMPATTs can be calculated from the knowledge of static spatial distribution of electron and hole ionization rates (αn(x) vs. x and αp(x) vs. x) obtained from the DC simulation, saturated drift velocities of charge carriers (vsn and vsp). If τAn and τAp are the avalanche response times initiated by electrons and holes, respectively, then these are expressed as (Acharyya and Banerjee 2012, 2013, Acharyya et al. 2012):

$$ \tau_{\text{An}} = \frac{1}{{\left( {v_{\text{sn}} + v_{\text{sp}} } \right)}}\int\limits_{{ - x_{{{\text{A}}_{1} }} }}^{{x_{{{\text{A}}_{2} }} }} {\exp \left[ { - \int\limits_{{ - x_{{{\text{A}}_{1} }} }}^{x} {\left( {\alpha_{\text{n}} - \alpha_{\text{p}} } \right)dx^{'} } } \right]} dx, $$
(11)
$$ \tau_{\text{Ap}} = \tau_{\text{An}} \exp \left[ {\int\limits_{{ - x_{{{\text{A}}_{1} }} }}^{{x_{{{\text{A}}_{2} }} }} {\left( {\alpha_{\text{n}} - \alpha_{\text{p}} } \right)dx} } \right]. $$
(12)

When avalanche process is initiated by a mixture of electrons and holes then the corresponding response time τA is given by (Acharyya and Banerjee 2012, 2013, Acharyya et al. 2012)

$$ \tau_{\text{A}} = \tau_{\text{An}} \left\{ {\left( {1 - k} \right) + k.\exp \left[ { - \int\limits_{{ - x_{{{\text{A}}_{1} }} }}^{{x_{{{\text{A}}_{2} }} }} {\left( {\alpha_{\text{n}} - \alpha_{\text{p}} } \right)dx} } \right]} \right\}^{ - 1} $$
(13)

where the parameter k = Jps/Js and (1 − k) = Jns/Js; Js = Jps + Jns is the total reverse saturation current of the device under dark condition. The expressions for thermally generated electron and hole reverse saturation currents are given by

$$ J_{\text{ns}} = \left[ {\frac{{qD_{\text{n}} n_{\text{i}}^{2} }}{{L_{\text{n}} N_{\text{A}} }}} \right] \quad {\text{and}}\quad J_{\text{ps}} = \left[ {\frac{{qD_{\text{p}} n_{\text{i}}^{2} }}{{L_{\text{p}} N_{\text{D}} }}} \right]. $$
(14)

High-frequency simulation

The magnitude of peak field at the junction (ξp), the widths of avalanche and drift zones (xA and xD; where xD = dn + dp) and the voltage drops across these zones (VA, VD) are obtained from the DC simulation program. These values are fed back as input parameters in the high-frequency simulation program. The depletion layer edges of the device are obtained from the output of DC simulation program. The edges of the depletion layer are then taken as the starting and end points of high-frequency simulation program. Two second-order differential equations are framed from Gummel–Blue model (Roy et al. 1985) by resolving the device impedance Z(x,ω) into its real part R(x,ω) and imaginary part X(x,ω); where Z(x,ω) = R(x,ω) + jX(x,ω). Two simultaneous second-order differential equations in R and X are numerically solved using Runge–Kutta method (Roy et al. 1985). Double-iteration over the initial choice of the values of R and X at one edge is carried out till the boundary conditions for R and X are satisfied at the other edge (Acharyya and Banerjee 2012; Roy et al. 1985). The negative specific resistance (R(x) vs. x) and specific reactance (X(x) vs. x) profiles in the depletion layer of the device are obtained from the above-mentioned solution. The device negative resistance (ZR) and reactance (ZX) are obtained from the numerical integration of the respective R(x)- and X(x)-profiles over the depletion layer width (W), i.e.,

$$ Z_{\text{R}} = \int\limits_{{ - x_{1} }}^{{x_{2} }} {R\left( x \right)dx} \quad {\text{and}}\quad Z_{\text{X}} = \int\limits_{{ - x_{1} }}^{{x_{2} }} {X\left( x \right)dx} . $$
(15)

The device impedance is given by ZD = ZR + jZX, while the device admittance is YD = Z −1D  = G + jB. The negative conductance (G) and positive susceptance (B) of the device at a particular frequency are computed from the following expressions:

$$ \left| {G\left( \omega \right)} \right| = \frac{{Z_{\text{R}} }}{{\left( {Z_{\text{R}}^{2} + Z_{\text{X}}^{2} } \right)}} \quad {\text{and}}\quad \left| {B\left( \omega \right)} \right| = \frac{{ - Z_{\text{X}} }}{{\left( {Z_{\text{R}}^{2} + Z_{\text{X}}^{2} } \right)}}. $$
(16)

It may be noted that both G and B are normalized with respect to the effective junction area (Aj) of the device. The admittance characteristics, i.e., G(ω) versus B(ω) plots of the device are obtained from the above analysis for different bias current densities. The avalanche resonance frequency (fa) at which the susceptance (B) changes its sign from positive to negative, i.e., from inductive to capacitive is also obtained from the admittance plots. The RF power output PRF from the device can be obtained from the following expression:

$$ P_{\text{RF}} = \frac{1}{2}\left( {V_{\text{RF}} } \right)^{2} \left| {G_{\text{p}} } \right|A_{\text{j}} , $$
(17)

where VRF is the RF voltage (VRF = mx × VB; where the modulation index, mx ≤ 30 % under small-signal condition; whereas mx is taken as 25 % in the calculations presented in this paper), |Gp| is the peak magnitude of the negative conductance at peak optimum frequency (fp) and Aj is the effective junction area of the device (Aj = π(Dj/2)2, and Dj is the effective junction diameter of the device (assuming the circular cross-section of the device)).

In the “Results and discussion” section, the DC and high-frequency results of DDR IMPATT devices based on different semiconductors are presented and discussed in detail. The numerical simulation approach described briefly in this section is verified by the authors by comparing the simulation obtained results of Si, GaAs and InP DDR IMPATTs presented in this paper at different mm-wave frequencies with published experimental reports. So far as authors’ knowledge is concerned no experimental report on 4H-SiC and Wz-GaN DDR IMPATTs is available in published literature within the mm-wave and THz frequency range under consideration. That is way the simulation predicted results of 4H-SiC and Wz-GaN DDR IMPATTs could not be verified.

Results and discussion

The active layer widths (Wn, Wp) and background doping concentrations (ND, NA) of DDR IMPATT devices (Table 1) based on different semiconductor materials have been designed to operate at different mm-wave and terahertz frequencies using simple transit time formula (Sze and Ryder 1971) and simulated using DC (Acharyya and Banerjee 2012; Roy et al. 1979) and high-frequency (Acharyya and Banerjee 2012; Roy et al. 1985) simulation techniques as described briefly in the previous section.

DC properties

Important DC parameters such as peak electric field (ξp), breakdown voltage (VB), avalanche zone voltage (VA), DC to RF conversion efficiency (η), avalanche layer width (xA) and ratio of avalanche zone width to total depletion layer width (xA/W) of the designed DDR IMPATTs are obtained from DC simulation and given in Table 2. Variations of ξp, VB and η with operating frequency of GaAs and Si-based DDR IMPATTs are shown in Fig. 2. Variations of the above parameters (ξp and VB, η) of InP, 4H-SiC and Wz-GaN-based DDR IMPATTs with operating frequency are shown in Figs. 3, 4. Thus, a comparison of the DC properties of DDR IMPATTs based on different semiconductors can be made from Table 2 and Figs. 2, 3, 4 at different operating frequencies. Table 2 shows that peak electric field (ξp) increases while the breakdown voltage (VB), avalanche zone voltage (VA), DC to RF conversion efficiency (η) and avalanche layer width (xA) decrease in the DDR IMPATTs based on different materials as the operating frequency increases. Peak electric field (ξp) is highest in 4H-SiC based DDRs among all other DDR IMPATTs under consideration at any operating frequency; it increases from 3.59 × 108 to 5.17 × 108 V m−1 as the operating frequency increases from 94 GHz to 1.0 THz. In Wz-GaN based DDR IMPATTs, ξp is of the same order (~108 V m−1) but lower than 4H-SiC DDRs. Peak electric fields (ξp) of GaAs, Si and InP-based DDR IMPATTs are one order lower (~107 V m−1) as compared with 4H-SiC and Wz-GaN based DDR IMPATTs. But among GaAs, Si and InP-based DDR IMPATTs, the peak field is highest in InP IMPATTs followed by Si and GaAs IMPATTs. Figure 4 and Table 2 show that the breakdown voltage (VB) of Wz-GaN based DDR IMPATTs is highest among all other semiconductor-based DDR devices up to 0.3 THz beyond which the breakdown voltage (VB) of 4H-SiC based IMPATTs exceeds that of Wz-GaN based IMPATTs. It is observed that GaAs, Si and InP-based DDRs have lower breakdown voltage (VB) than either 4H-SiC or Wz-GaN based DDRs. It is observed from Table 2 as well as Figs. 2 and 4 that breakdown voltage (VB) of InP DDR device is highest among GaAs, Si and InP DDRs at the same operating frequencies. From the knowledge of avalanche zone voltage (VA) and breakdown voltage (VB), the drift zone voltage drop (VD = VB − VA) of the device can be calculated. The ratio of drift zone voltage to breakdown voltage (VD/VB) decreases in all the devices under consideration as the operating frequency increases. The ratio VD/VB is much higher in Wz-GaN DDRs (0.510 at 94 GHz, 0.419 at 1.0 THz) and 4H-SiC DDRs (0.509 at 94 GHz, 0.295 at 1.0 THz) as compared with GaAs DDRs (0.313 at 94 GHz, 0.183 at 220 GHz) or Si DDRs (0.332 at 94 GHz, 0.145 at 0.5 THz) and InP DDRs (0.489 at 94 GHz, 0.246 at 1.0 THz). The magnitudes of VD/VB in Wz-GaN, 4H-SiC and InP DDRs are comparable up to lower mm-wave frequencies. At higher frequencies the ratio VD/VB decreases sharply in 4H-SiC and InP DDRs. It may be noted that the rate of decrease of VD/VB with respect to frequency is much less in Wz-GaN based DDR IMPATTs. The ratio VD/VB is nearly same in GaAs and Si DDRs up to some limiting frequency beyond which VD/VB is higher in Si DDRs. According to the semi-quantitative formula of DC to RF conversion efficiency (η = (1/π) × (VD/VB)) (Scharfetter and Gummel 1969) the DC to RF conversion efficiency (η) of IMPATT devices is directly proportional to the ratio VD/VB. Thus, the variation of DC to RF conversion efficiency (η) of the devices with frequency exhibits the same nature as that of VD/VB. Figure 4 and Table 2 show that the DC to RF conversion efficiencies (η) of InP, 4H-SiC and Wz-GaN DDRs are comparable and much greater than DC to RF conversion efficiencies (η) of GaAs and Si DDRs as shown in Fig. 2 and Table 2. DC to RF conversion efficiency (η) is highest in Wz-GaN DDRs and rate of decrease of η with respect to operating frequency (dη/df) is also much lower (Fig. 4) in Wz-GaN DDRs than that in InP and 4H-SiC DDRs up to 1.0 THz.

Table 2 DC properties
Full size table
Fig. 2
figure 2

Variations of DC to RF conversion efficiency, breakdown voltage and peak electric field with optimum frequency of GaAs and Si-based DDR IMPATTs

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Fig. 3
figure 3

Variation of peak electric field with optimum frequency of InP, 4H-SiC and Wz-GaN based DDR IMPATTs

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Fig. 4
figure 4

Variations of DC to RF conversion efficiency and breakdown voltage with optimum frequency of InP, 4H-SiC and Wz-GaN based DDR IMPATTs

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The ratio of avalanche zone width to total drift layer width (xA/W) for all the devices under consideration increases sharply at higher operating frequencies. Higher xA/W indicates wider avalanche zone, which leads to higher avalanche voltage (VA) and lower drift zone voltage (VD). The lower the VD/VB, the lower the efficiency (η). Thus, the rapid widening of avalanche region at higher operating frequencies is the primary cause of sharp decrease of η at higher operating frequencies. In case of GaA-based DDRs xA/W is 40.0 % at 60 GHz but it rises to 66.36 % at 220 GHz which causes sharp decrease of η at 220 GHz frequency. But in Si DDRs at 220 GHz, xA/W is 53.10 %, but it rises to 67.16 % at 500 GHz and this leads to sharp fall in η at higher mm-wave and THz frequencies. In case of InP, 4H-SiC and Wz-GaN DDR devices the magnitude of xA/W is much lower up to a frequency of 0.3 THz (41.79 % in InP DDR, 37.08 % in 4H-SiC DDR, 38.39 % in Wz-GaN), but it increases rapidly above 0.3 THz in InP and 4H-SiC DDRs and becomes 67.85 and 61.83 % at 1.0 THz, respectively. This leads to very low value of η (7.82 and 9.38 % respectively) for InP and 4H-SiC DDRs at that frequency. But in Wz-GaN DDRs, sharp rise in xA/W is observed at frequencies above 1.0 THz whose magnitude increases up to 67.69 % at 5.0 THz leading to very low efficiency (5.58 %) of Wz-GaN DDRs at that frequency.

Avalanche response times

Avalanche response times (τA) of the DDR IMPATTs based on different semiconductors are calculated by Eq. (13). Spatial variations of ionization rates of electrons (αn) and holes (αp) for each device are obtained from the output of the DC program and used to solve Eqs. (11) and (13). Transit times (τT) associated with the DDR IMPATT devices based on GaAs, Si, InP, 4H-SiC and Wz-GaN operating at different mm-wave and THz frequencies are obtained from transit time formula of Sze and Ryder (1971). The computed values of τA and τT of all DDR devices under consideration are given in Table 3. Figures 5 and 6 show the variations of τA and τT with operating frequency in GaAs, Si-based DDRs and InP, 4H-SiC, Wz-GaN based DDRs, respectively. DDR IMPATTs based on a particular semiconductor can be operated at higher frequencies provided the avalanche response time (τA) is much lower than the transit time (τT) of carriers at the higher frequencies (Acharyya and Banerjee 2012, 2013). Avalanche response time (τA) of the device depends on the saturated drift velocities and ionization rates of electron and holes (vsn, vsp and αn, αp, respectively) of the base semiconductor. Thus, the type of the base material determines the value of τA at a particular frequency. Figure 5 and Table 3 show that τA of GaAs DDRs is higher than that of Si DDRs (at 94 GHz τA of GaAs DDR is 1.72 times higher than τA of Si DDR; whereas at 220 GHz this ratio is 2.15). That is why DDR IMPATTs based on Si can operate at much higher frequencies (up to 0.5 THz) than those based on GaAs (up to 220 GHz). It is observed from Table 3 that the values of τA of InP, 4H-SiC and Wz-GaN DDRs are lower than those of GaAs and Si DDRs. Further from Fig. 6 and Table 3, it is observed that τA of Wz-GaN DDR IMPATTs is much smaller than that of InP DDR IMPATTs. At 94 GHz, τA of Wz-GaN DDR is 9.18 × 10−4 times of InP DDR and τA of 4H-SiC DDR is 6.26 × 10−3 times of InP DDR at the same frequency, whereas the same ratio of τA of Wz-GaN DDR to τA of InP and 4H-SiC DDRs become 2.82 × 10−4 and 8.58 × 10−3, respectively, at 1.0 THz. It therefore turns out that τA of Wz-GaN DDRs is the lowest of all other DDR IMPATTs under consideration (at 94 GHz τA of Wz-GaN DDR is 0.15 times than that of 4H-SiC DDR, whereas at 1.0 THz this ratio becomes 0.33). Due to very small avalanche response time (τA), of the order of ~10−16 s, Wz-GaN DDR IMPATT device is capable of operating up to 5.0 THz, whereas the highest operating frequencies of GaAs, Si, InP and 4H-SiC DDRs are limited to 220 GHz, 500 GHz, 1.0 THz and 1.0 THz, respectively.

Table 3 Avalanche response times and transit times
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Fig. 5
figure 5

Variations of avalanche response time and transit time with optimum frequency of GaAs and Si-based DDR IMPATTs

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Fig. 6
figure 6

Variations of avalanche response time and transit time with optimum frequency of InP, 4H-SiC and Wz-GaN based DDR IMPATTs

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High-frequency properties

The important high-frequency parameters obtained from this analysis are optimum frequency (fp), peak negative conductance (Gp), corresponding susceptance (Bp), quality factor or Q-factor (Qp), negative resistance (ZR) and RF power output (PRF). Those parameters are obtained from the high-frequency simulation of GaAs, Si, InP, 4H-SiC and Wz-GaN DDR IMPATTs and listed in Table 4. Admittance characteristics or high-frequency conductance-susceptance plots of GaAs and Si DDRs are shown in Fig. 7 while those of InP and 4H-SiC DDRs are shown in Figs. 8 and 9, respectively, and those of Wz-GaN DDRs are shown in Figs. 10 and 11, respectively. It is observed from Table 4 and Figs. 7, 8, 9, 10, 11 that the magnitudes of Gp and Bp increase with the increase of operating frequency. Q-factor (Qp = −Bp/Gp) of the device determines the growth rate of IMPATT oscillation. Lower Q-factor close to Qp ≈ 1 suggests higher oscillation growth rate. It is observed from Table 4 that the Q-factors of 4H-SiC DDRs are lowest (Qp < 1) from 94 GHz to 1.0 THz of all other DDRs under consideration. It is interesting to note that Q-factor (Qp) of GaAs, Si and InP DDRs increases with the increase of operating frequency, but Q-factor (Qp) of Wz-GaN DDR IMPATTs decreases with the increase of operating frequency. This fact suggests that the oscillation growth rate improves in Wz-GaN based DDRs at higher frequencies (up to 5.0 THz).

Table 4 High-frequency properties
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Fig. 7
figure 7

Admittance characteristics of GaAs and Si-based DDR IMPATTs

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Fig. 8
figure 8

Admittance characteristics of 94, 140, 220 GHz, 0.3 THz InP and 4H-SiC based DDR IMPATTs

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Fig. 9
figure 9

Admittance characteristics of 0.5, 1.0 THz InP and 4H-SiC based DDR IMPATTs

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Fig. 10
figure 10

Admittance characteristics of 94, 140, 220 GHz, 0.3 THz Wz-GaN based DDR IMPATTs

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Fig. 11
figure 11

Admittance characteristics of 0.5, 1.0, 1.5, 2.0, 5.0 THz Wz-GaN based DDR IMPATTs

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Negative resistances (ZR) of the devices are calculated from Eq. (15) and given in Table 4. Table 4 shows that, the magnitude of negative resistance (ZR) of DDR IMPATTs based on GaAs, Si, InP, 4H-SiC and Wz-GaN decreases with the increase of operating frequency. It is interesting to observe that the magnitude of ZR is highest in 4H-SiC based DDRs among all other devices from 94 GHz to 1.0 THz. Higher magnitude of negative resistance implies that higher RF power output can be obtained from the 4H-SiC based DDR IMPATT devices in comparison with GaAs, Si, InP and Wz-GaN based DDR IMPATT devices operating at same frequency.

RF power outputs (PRF) of the DDR IMPATT devices under consideration at different operating frequencies are obtained from Eq. (17) from the knowledge of breakdown voltage (VB), voltage modulation factor (mx = 25 %), magnitude of peak negative conductance (Gp) and effective junction area of the device (Aj). The effective junction diameter (Dj) is scaled down from 55 to 0.7 μm as the frequency of operation increases from 60 GHz to 5.0 THz through a rigorous thermal analysis (Acharyya et al. 2011, 2010) in continuous-wave (CW) mode considering proper heat sinking aspects which is sufficient to avoid the thermal runway and burn out of the device and corresponding Dj values are given in Table 4. Variation of RF power output (PRF) with optimum frequency of GaAs, Si, InP, 4H-SiC and Wz-GaN-based DDR IMPATTs is shown in Fig. 12. Figure 12 and Table 4 show that PRF decreases as the operating frequency of the device increases. It is interesting to observe from Fig. 12 and Table 4 that RF power output (PRF) of GaAs and Si-based DDR IMPATTs is almost comparable up to 220 GHz; up to 94 GHz, GaAs DDRs can deliver marginally higher power output as compared with Si DDRs, but above 94 GHz Si DDRs exceed GaAs DDRs as regards PRF. It is also observed that InP-based DDR IMPATTs are capable of delivering higher output power as compared with DDR IMPATTs based on GaAs and Si. Another advantage of InP DDR IMPATTs is that the device can operate up to much higher frequencies (up to 1.0 THz), whereas the high-frequency performance of GaAs and Si DDR IMPATTs are limited to 220 GHz and 0.5 THz, respectively. It is interesting to note that Wz-GaN and 4H-SiC DDR IMPATTs are potential candidates for THz operation due to their higher RF power outputs at THz frequencies. Up to 1.0 THz 4H-SiC IMPATTs provide the highest RF power output among all other devices under consideration. But the genuine advantage of Wz-GaN DDR IMPATTs is that it can operate up to 5.0 THz with appreciable power output.

Fig. 12
figure 12

Variation of RF power output with optimum frequency of GaAs, Si, InP, 4H-SiC and Wz-GaN based DDR IMPATTs

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Validation of the simulation results

Experimentally obtained RF power output from Si DDR IMPATT sources are 600 mW at 94 GHz (Luy et al. 1987) and 300 mW at 140 GHz (Wollitzer et al. 1996) which are shown in Fig. 12. Figure 12 shows that the experimental results are in very close agreement with the simulation results for Si DDR IMPATTs which validates the DC and high-frequency simulation technique adopted by the authors and used in this paper. Experimentally obtained RF power output from GaAs DDR IMPATT sources are 1.24 W, 270 and 100 mW at 60, 95 and 144 GHz, respectively (Adlerstein and Chu 1984; Eisele 1989; Tschernitz and Freyer 1995), and the same from InP DDR IMPATT source is 55 mW at 84.8 GHz (Eisele et al. 1996). Experimentally obtained power outputs from GaAs and InP IMPATT sources are also shown in Fig. 12. It can be observed from Fig. 12 that at 60 GHz the simulated and experimental results for GaAs DDR IMPATT source are in close agreement; but above 60 GHz simulation results are predicting much higher power output as compared with the experimentally obtained values. Similarly, at W-band (75–110 GHz) simulation predicted RF power out is much higher as compared with the experimentally obtained value for InP DDR IMPATT source. Lack of mature fabrication technology of GaAs and InP-based DDR IMPATT devices higher mm-wave frequency bands and lack of optimized design of the devices used for those experiments may be the cause of getting lower power outputs experimentally. But so far as the authors’ knowledge is concerned, no experimental report on 4H-SiC and Wz-GaN based DDR IMPATT source operating at mm-wave or THz frequency bands is available in published literatures. That is why the simulation results of 4H-SiC and Wz-GaN based DDR IMPATTs presented in this paper could not be compared with experimentally obtained results.

Conclusions

Millimeter-wave and Terahertz performance of DDR IMPATT devices based on GaAs, Si, InP, 4H-SiC and Wz-GaN are investigated in this paper. High-frequency limitations of DDR IMPATTs based on the above-mentioned semiconductors are studied from the avalanche response time of the devices operating at mm-wave and THz frequencies. Wz-GaN DDR IMPATTs emerge as the most suitable material for generation of THz frequencies up to 5.0 THz due to their small avalanche response time, high DC to RF conversion efficiency and sufficiently high RF power output of Wz-GaN based DDRs at THz frequencies. But up to 1.0 THz, 4H-SiC DDR IMPATTs excel Wz-GaN DDR IMPATTs due to their higher power output. It may therefore be concluded that Wz-GaN and 4H-SiC are highly suitable semiconductors for IMPATT action in both mm-wave and THz frequency ranges with respect to power output and efficiency. The simulation results and corresponding designs would be useful for undertaking an experimental program to fabricate the THz 4H-SiC and Wz-GaN IMPATTs using MBE or metallo-organic chemical vapor deposition techniques.